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Abstract: Currently there is an increasing research activity in the area of climate extremes because they represent a key manifestation of non-linear systems and an enormous impact on economic and social human activities. Our understanding of the mean behavior of climate and its ‘normal’ variability has been improving significantly during the last decades. In comparison, climate extreme events have been hard to study and even harder to predict because they are, by definition, rare and obey different statistical laws than averages. In this context, the motivation for this paper is twofold. Firstly, we recall the basic principles of Extreme Value Theory that is used on a regular basis in finance and hydrology, but it still does not have the same success in climate studies. More precisely, the theoretical distributions of maxima and large peaks are recalled. The parameters of such distributions are estimated with the maximum likelihood estimation procedure that offers the flexibility to take into account explanatory variables in our analysis. Secondly, we detail three case-studies to show that this theory can provide a solid statistical foundation, specially when assessing the uncertainty associated with extreme events in a wide range of applications linked to the study of our climate. To cite this article: P. Naveau et al., C. R. Geoscience 337 (2005). [Copyright &y& Elsevier]